Related papers: $3$-Selmer group, ideal class groups and cube sum …

We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of…

We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the…

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers…

Let K be a multiquadratic number field. We investigate the average dimension of 2-Selmer groups over K for the family of all elliptic curves over the rational numbers (ordered by height). We give upper and lower bounds for this average. In…

Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log…

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable…

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in…

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…

A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian…

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which…

In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq…

Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for…

Let $E$ be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over $\mathbb{Q}$ constructed from $E$ through multiplicative induction; it is of rank $8$. We show that, under certain assumptions…

It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is…

We exhibit several families of elliptic curves with torsion group isomorphic to $ \Z/6\Z$ and generic rank at least $3$. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We…

By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may…

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all…

Consider a function field $K$ with characteristic $p>0$. We investigate the $\Lambda$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. Next, we study…

this paper is devoted to the study of curves of genus 3 with group of automorphisms the symmetric group S3, principally over finite fields, in view to obtain optimal curves. For instance, we prove that, over the finite fields of char. 3,…